Arithmetic Torelli maps for cubic surfaces and threefolds

TL;DR

This paper extends the construction of associated abelian varieties from complex cubic surfaces and threefolds to an arithmetic setting, addressing longstanding questions and classifying their Mumford-Tate groups.

Contribution

It provides a new arithmetic construction of Torelli maps for cubics and classifies the Mumford-Tate groups of the resulting abelian varieties.

Findings

Construction works over arithmetic bases, excluding prime 2

Answers old questions of Deligne and Kudla-Rapoport

Classifies Mumford-Tate groups of associated abelian varieties

Abstract

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the prime 2, an old question of Deligne and a recent question of Kudla and Rapoport. We further classify the Mumford-Tate groups of the abelian varieties which arise, and give additional arithmetic applications.